Proving the Dot Product Identity for Vector Fields and Their Curl

In summary, the student is trying to solve a problem in vector calculus and seems to be stuck. They have an equation involving the cross product, but don't seem to be able to solve for the perpendicular vector. They are also trying to figure out if a certain vector is perpendicular to another vector.
  • #1
Mario Carcamo
7
0

Homework Statement


http://faculty.fiu.edu/~maxwello/phz3113/probs/set1.pdf

I'm working on problem 2. Trying to prove that the dot product between a vector field and its curl is zero.

Homework Equations


The basic identities of vector calculus and how scalar fields and vector fields interact

The Attempt at a Solution


My only real attempt is expanding what was given using an identity. What i have now is that

f(del X A) = A X del(f)

In my head the proof is trivial since by the very definition of the cross product, the new vector while be perpendicular to both vectors that served as the argument. I do have a question though. In the above notation and in the identity when they say f(del X A) do they mean to input the curl vector as an argument for the scalar function? Thanks!
 
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  • #2
Mario Carcamo said:
In my head the proof is trivial since by the very definition of the cross product, the new vector while be perpendicular to both vectors that served as the argument.
I don't precisely know which vectors you were referring to by "both vectors", but I have a feeling that you are concerned with ##\mathbf{A}## being always perpendicular to ##\nabla \times \mathbf{A}## because the cross product in between the last expression makes it perpendicular to ##\mathbf{A}##. I believe this is not necessarily the case. Anyway, the more elegant way is to multiply by dot product the identity you have there with ##\mathbf{A}##.
Mario Carcamo said:
In the above notation and in the identity when they say f(del X A) do they mean to input the curl vector as an argument for the scalar function? Thanks!
No, they are just vector multiplied with a scalar.
 
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Likes Mario Carcamo
  • #3
blue_leaf77 said:
I don't precisely know which vectors you were referring to by "both vectors", but I have a feeling that you are concerned with ##\mathbf{A}## being always perpendicular to ##\nabla \times \mathbf{A}## because the cross product in between the last expression makes it perpendicular to ##\mathbf{A}##. I believe this is not necessarily the case. Anyway, the more elegant way is to multiply by dot product the identity you have there with ##\mathbf{A}##.

No, they are just vector multiplied with a scalar.

Yeah what i mean is if A X B = C then C must be perpendicular to A and B so if your doing Del X A = C then C must be perpendicular to A but maybe that is not the case when your talking about del?
 
  • #4
Mario Carcamo said:
but maybe that is not the case when your talking about del?
One of the identities involving curl is ##\nabla \times (f\mathbf{A}) + \mathbf{A}\times \nabla f= f(\nabla \times \mathbf{A}) ##. Try multiplying by dot product both sides with ##\mathbf{A}##. An example in electromagnetism is one of the Maxwell equations, ##\nabla \times \mathbf{E} = -\partial \mathbf{B} /\partial t##, in most cases electric and magnetic fields are perpendicular, except for certain cases like that in the wave propagation in waveguides (for example see http://physics.stackexchange.com/qu...ric-and-magnetic-fields-are-not-perpendicular).
 
  • #5
blue_leaf77 said:
One of the identities involving curl is ##\nabla \times (f\mathbf{A}) + \mathbf{A}\times \nabla f= f(\nabla \times \mathbf{A}) ##. Try multiplying by dot product both sides with ##\mathbf{A}##. An example in electromagnetism is one of the Maxwell equations, ##\nabla \times \mathbf{E} = -\partial \mathbf{B} /\partial t##, in most cases electric and magnetic fields are perpendicular, except for certain cases like that in the wave propagation in waveguides (for example see http://physics.stackexchange.com/qu...ric-and-magnetic-fields-are-not-perpendicular).

yeah i got the answer thanks alot!
 

Related to Proving the Dot Product Identity for Vector Fields and Their Curl

1. What is vector calculus?

Vector calculus is a branch of mathematics that deals with the study of vectors and vector fields, and their relationships with derivatives and integrals. It is used to analyze and describe physical quantities that have both magnitude and direction.

2. What are the basic operations in vector calculus?

The basic operations in vector calculus include vector addition, scalar multiplication, dot product, cross product, and differentiation and integration of vector functions.

3. How is vector calculus used in real life applications?

Vector calculus is used in many fields of science and engineering, such as physics, engineering, economics, and computer graphics. It is used to model and analyze various physical phenomena, such as motion, forces, and electromagnetic fields.

4. What are the key concepts in vector calculus?

The key concepts in vector calculus include vector fields, line integrals, surface integrals, and the fundamental theorem of calculus for line and surface integrals. Other important concepts include vector derivatives, gradient, divergence, and curl.

5. How can I improve my understanding of vector calculus?

To improve your understanding of vector calculus, it is important to practice solving problems and familiarize yourself with the key concepts and formulas. You can also seek help from a tutor or attend a study group to clarify any doubts or difficulties you may have.

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